Method of correcting overlay and semiconductor device manufacturing method using the same
Summary by NHIP
Overlay correction using Zernike polynomials
The method calculates overlay correction values by applying measurement data to a polar coordinate function. This function utilizes a Zernike polynomial containing specific terms such as 1, R cos θ, and R sin θ, where the C 0 term represents an offset deformation value.
Claim Score by NHIP
Abstract
A method of correcting an overlay includes setting a reference map having information relating to predetermined positions of a substrate. An overlay value is measured at each of the predetermined positions to obtain a plurality of overlay measurement values. The plurality of overlay measurement values is applied to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function. The polar coordinate function uses coordinate values of the predetermined positions as parameters.
Term
4.8 yearsleft in the term
Expires 7 July 2031, including 140 days of term adjustment.
- Priority
- Filed
- Granted
- Today
- Expires
7 claims: 4 independent, 3 dependent
- 1A method of correcting an overlay, comprising:setting a reference map having information relating to predetermined positions of a substrate;measuring an overlay value at each of the predetermined positions to obtain a plurality of overlay measurement values;applying the plurality of overlay measurement values to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function, wherein parameters of the polar coordinate function include coordinate values of the predetermined positions;applying the correlation coefficient to the polar coordinate function with each corresponding coordinate value in substantially the reference map being a parameter to calculate an overlay correction value for the corresponding coordinate value;and generating a correction map such that corresponding positions of the reference map are corrected with corresponding overlay correction values, wherein the polar coordinate function comprises a Zernike polynomial including a plurality of terms of 1, R cos θ, R sin θ, and (2R 2 −1) and the product of the Zernike polynomial and the correlation coefficient including at least C 0 , and wherein the C 0 corresponds to an offset deformation value of the plurality of overlay measurement values, R corresponds to a radial component, and θ corresponds to an azimuth angle component.
- 3A method of correcting overlay, comprising:setting a reference map having information relating to predetermined positions of substrate;measuring an overlay value at each of the predetermined positions to obtain a plurality of overlay measurement values;applying the plurality of overlay measurement values to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function, wherein parameters of the polar coordinate function include coordinate values of the predetermined positions;applying the correlation coefficient to the polar coordinate function with, each corresponding coordinate value in substantially the entire reference map being a parameter, to calculate an overlay correction value for each corresponding coordinate value;and generating a correction map such that corresponding positions of the reference map are corrected with corresponding overlay correction values, wherein the polar coordinate function comprises a Zernike polynomial including a plurality of terms of 1, R cos θ, R sin θ, and (2R 2 −1) and the product of the Zernike polynomial and the correlation coefficient including at least C 2 R cos θ+C 4 R sin θ;and wherein the C 2 and C 4 correspond to a scale deformation value of the plurality of overlay measurement values, R corresponds to a radial component, and θ corresponds to an azimuth angle component.
- 4A method of correcting overlay comprising:setting a reference map having information relating to predetermined positions of a substrate;measuring an overlay value at each of the predetermined positions to obtain a plurality of overlay measurement values;applying the plurality of overlay measurement values to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function, wherein parameters of the polar coordinate function include coordinate values of the predetermined positions;applying the correlation coefficient to the polar coordinate function with, each corresponding coordinate value in substantially the entire reference map being a parameter, to calculate an overlay correction value for each corresponding coordinate value;and generating a correction map such that corresponding positions of the reference map are corrected with corresponding overlay correction values, wherein the polar coordinate function com rises a Zernike polynomial including a plurality of terms of 1, R cos θ, R sin θ, and (2R 2 −1) and the product of the Zernike polynomial and the correlation coefficient including at least C 6 (2R 2 −1);and wherein the C 6 corresponds to a high-order deformation value, R corresponds to a radial component, and θ corresponds to an azimuth angle component.
- 5Broadest claimClaim Score 41, average(NHIP)A method of manufacturing a semiconductor device, the method comprising:forming a lower pattern on a substrate in a unit process equipment;forming an upper pattern on the lower pattern with a reference map in an exposure equipment;and correcting an overlay, the correcting the overlay including: setting the reference map having information relating to predetermined positions of the substrate;measuring an overlay value for each of the predetermined positions to obtain a plurality of overlay measurement values corresponding to a distortion degree between the lower pattern and the upper pattern;and applying the plurality of overlay measurement values to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function, wherein parameters of the polar coordinate function include coordinate values of the predetermined positions and wherein the overlay further includes: applying the correlation coefficient to the polar coordinate function, with each corresponding coordinate value in substantially the entire reference map being a parameter, to calculate an overlay correction value for each corresponding coordinate value, wherein correcting the overlay further includes generating a correction map such that corresponding positions of the reference map are corrected with corresponding overlay correction values.
Independent claims4
80 paragraphs in 4 sections, as filed
BACKGROUND
p-00021. Field
p-0003Embodiments relates to a semiconductor device manufacturing method, and more particularly, to a method of correcting an overlay value between two different layers and a semiconductor device manufacturing method using the same.
p-00042. Description of the Related Art
p-0005An overlay measuring process may be used to detect a distortion degree between a lower pattern and an upper pattern formed on a substrate. The overlay measuring process may be performed at a plurality of positions on the substrate. Thus, a plurality of overlay measurement values can be obtained on a single substrate. The plurality of overlay measurement values may be different from each other.
p-0006Methods of correcting an overlay; however, may exhibit a disadvantage in that reliability of the overlay measuring process is deteriorated based on a lack of improvement in the uniformity of the several overlay measurement values.
SUMMARY
p-0007Embodiments are directed to a method of correcting an overlay value.
p-0008Embodiments are also directed to a method of fabricating a semiconductor device using the method of correcting an overlay value.
p-0009Exemplary embodiments provide a method of correcting an overlay. The method of correcting the overlay includes setting a reference map having information relating to predetermined positions of a substrate. An overlay value is measured at each of the predetermined positions to obtain a plurality of overlay measurement values. The plurality of overlay measurement values is applied to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function. The polar coordinate function uses coordinate values of the predetermined positions as parameters.
p-0010In some embodiments, the polar coordinate function may include a Zernike polynomial.
p-0011In other embodiments, the Zernike polynomial may include a plurality of terms of 1, R cos θ, R sin θ, (2R<sup>2</sup>−1), R<sup>2 </sup>cos θ, R<sup>2 </sup>sin θ, (3R<sup>3</sup>−2R)cos θ, (3R<sup>3</sup>−2R)sin θ, (6R<sup>4</sup>−6R<sup>2</sup>+1), R<sup>3 </sup>cos 3θ, R<sup>3 </sup>sin 3θ, (4R<sup>4</sup>−3R<sup>2</sup>)cos 2θ, (4R<sup>4</sup>−3R<sup>2</sup>)sin 2θ, (10R<sup>5</sup>−12R<sup>3</sup>+3R)cos θ, (10R<sup>5</sup>−12R<sup>3</sup>+3R)sin θ, 20R<sup>6</sup>−30R<sup>4</sup>+12R<sup>2</sup>−1, R<sup>4 </sup>cos 4θ, R<sup>4 </sup>sin 4θ, (5R<sup>5</sup>+4R<sup>3</sup>)cos 3θ, (5R<sup>5</sup>+4R<sup>3</sup>)sin 3θ, (15R<sup>6</sup>−20R<sup>4</sup>+6R<sup>2</sup>)cos 2θ, (15R<sup>6</sup>−20R<sup>4</sup>+6R<sup>2</sup>)sin 2θ, (35R<sup>7</sup>−60R<sup>5</sup>+30R<sup>3</sup>−4R)cos θ, (35R<sup>7</sup>−60R<sup>5</sup>+30R<sup>3</sup>−4R)sin θ, and 70R<sup>8</sup>−140R<sup>6</sup>+90R<sup>4</sup>−20R<sup>2</sup>+1.
p-0012In still other embodiments, the polar coordinate function may include the product of the Zernike polynomial and the correlation coefficient.
p-0013In even other embodiments, the product of the Zernike polynomial and the correlation coefficient may include C<sub>0</sub>+C<sub>2</sub>R cos θ+C<sub>4</sub>R sin θ+C<sub>6</sub>(2R<sup>2</sup>−1)+ . . . .
p-0014In yet other embodiments, the C<sub>0 </sub>may correspond to an offset deformation value of the overlay measurement values.
p-0015In further embodiments, the C<sub>2 </sub>and C<sub>4 </sub>may correspond to a scale deformation value of the overlay measurement values.
p-0016In still further embodiments, the C<sub>6 </sub>may correspond to a high-order deformation value.
p-0017In even further embodiments, the product of the Zernike polynomial and the correlation coefficient may include the product of a Zernike polynomial matrix and a correlation coefficient matrix.
p-0018In yet further embodiments, the product of the Zernike polynomial matrix and the correlation coefficient matrix may include:
p-0019<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>C</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></math></maths>
p-0020In yet further embodiments, the product of the Zernike polynomial matrix and the correlation coefficient matrix may correspond to a matrix of the plurality of overlay measurement values.
p-0021In yet further embodiments, the matrix of the plurality of overlay measurement values may include:
p-0022<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mrow><mo> </mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>dx</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></math></maths>
p-0023In yet further embodiments, the correlation coefficient matrix may correspond to the product of a Zernike polynomial transposed matrix and the matrix of the plurality of overlay measurement values.
p-0024In yet further embodiments, the product of a Zernike polynomial transposed matrix and the matrix of the plurality of overlay measurement values may include:
p-0025<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>…</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow><mo></mo><mrow><mo> </mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>dx</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></math></maths>
p-0026In yet further embodiments, the method may further include multiplying the correlation coefficient matrix by the Zernike polynomial matrix to calculate an overlay correction value corresponding to each of related position coordinate values in the entire surface of the substrate. The Zernike polynomial matrix uses each of the related position coordinate values as a parameter.
p-0027In yet further embodiments, the method may further include generating a correction map by updating the reference map with the overlay correction value.
p-0028Exemplary embodiments also provide a method of manufacturing a semiconductor device. The method of manufacturing the semiconductor device includes forming a lower pattern on a substrate in a unit process equipment, forming an upper pattern on the lower pattern by using a reference map in an exposure equipment, and correcting an overlay. Correcting the overlay includes setting a reference map having information relating to predetermined positions of a substrate. An overlay value is measured at each of the predetermined positions to obtain a plurality of overlay measurement values corresponding to a distortion degree between the lower pattern and the upper pattern. The plurality of overlay measurement values is applied to a polar coordinate function to calculate a correlation coefficient of the polar coordinate function. The polar coordinate function uses coordinate values of the predetermined positions as parameters.
p-0029In some embodiments, the polar coordinate function may include a Zernike polynomial.
p-0030In other embodiments, correcting the overlay may further include applying the correlation coefficient to the polar coordinate function using each of related position coordinate values in the entire reference map as a parameter to calculate an overlay correction value in the related position coordinate value, and generating a correction map of which the related positions of the reference map are corrected with an overlap correction value.
p-0031In still other embodiments, correcting the overlay may further include sending the correction map to the exposure equipment in the overlay measuring equipment.
BRIEF DESCRIPTION OF THE DRAWINGS
p-0032Features will become more apparent to those of ordinary skill in the art by describing in detail exemplary embodiments with reference to the attached drawings, in which:
p-0033<figref idrefs="DRAWINGS">FIG. 1</figref> illustrates a schematic diagram of a semiconductor device manufacturing equipment used in a method of correcting an overlay according to an exemplary embodiment;
p-0034<figref idrefs="DRAWINGS">FIG. 2</figref> illustrates a flowchart of a method of fabricating a semiconductor device according to an exemplary embodiment; and
p-0035<figref idrefs="DRAWINGS">FIGS. 3 through 5</figref> illustrate plan views of a method of fabricating a semiconductor device according to an exemplary embodiment.
DETAILED DESCRIPTION
p-0036Korean Patent Application No. 10-2010-0015312, filed on Feb. 19, 2010, in the Korean Intellectual Property Office, and entitled: “Method of Correcting Overlay and Semiconductor Device Manufacturing Method Using the Same,” is incorporated by reference herein in its entirety.
p-0037Hereinafter, exemplary embodiments will be described in detail with reference to the accompanying drawings. Features of the present invention and methods of accomplishing the same may be understood more readily by reference to the following detailed description of preferred embodiments and the accompanying drawings. The embodiments may, however, be embodied in many different forms and should not be construed as being limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the scope of the inventive concept to those skilled in the art, and the embodiments of the inventive concept will only be defined by the appended claims. Like reference numerals refer to like elements throughout the specification.
p-0038The terminology used herein is for the purpose of describing various embodiments only and is not intended to be limiting of example embodiments. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated elements, steps, operations, and/or components, but do not preclude the presence or addition of one or more other elements, steps, operations, components, and/or groups thereof.
p-0039<figref idrefs="DRAWINGS">FIG. 1</figref> is a schematic diagram of a semiconductor device manufacturing equipment used in a method of correcting an overlay according to an exemplary embodiment.
p-0040Referring to <figref idrefs="DRAWINGS">FIG. 1</figref>, a semiconductor device manufacturing equipment <b>100</b> may include at least one overlay measuring equipment <b>40</b> and at least one exposure equipment <b>30</b>. The overlay measuring equipment <b>40</b> measures at least one overlay value which may correspond to a distortion degree between a lower pattern on a substrate and a photoresist pattern on the lower pattern. The distortion degree may comprise a misalignment value between the lower pattern and the photoresist pattern. The exposure equipment <b>30</b> may have a function to selectively expose some portions of a photoresist layer using a mask pattern. The overlay measuring equipment <b>40</b> may calculate an overlay correction value using the measured overlay value and may generate a correction map using the overlay correction value. The exposure equipment <b>30</b> may receive the overlay correction value and the correction map, thereby performing a new exposure process to reflect the correction map.
p-0041The overlay measuring equipment <b>40</b> may communicate data such as the overlay measurement value and the overlay correction value calculated from the overlay measurement value with the exposure equipment <b>30</b> via a host computer <b>10</b> or a server. The overlay measuring equipment <b>40</b> and the exposure equipment <b>30</b> may have information regarding a reference map. The reference map may include at least one coordinate indicating at least one reference position of the substrate. Moreover, the overlay measuring equipment <b>40</b> and the exposure equipment <b>30</b> may update the reference map to make the overlay correction value and may generate the correction map.
p-0042The semiconductor device manufacturing equipment <b>100</b> may further include at least one spinner equipment <b>20</b>. The spinner equipment <b>20</b> may have a function to form the photoresist layer on the substrate. Moreover, the spinner equipment <b>20</b> may have another function to develop the photoresist layer which is exposed by the exposure equipment <b>30</b>. As a result, the photoresist pattern is finally formed by the spinner equipment <b>20</b>. The overlay measuring equipment <b>40</b> may perform an overlay measuring process to measure the overlay value between the lower pattern and the photoresist pattern using an optical microscopy.
p-0043The semiconductor device manufacturing equipment <b>100</b> may further comprise at least one unit process equipment <b>50</b>. The unit process equipment <b>50</b> may comprise at least one of various apparatus. For example, the unit process equipment <b>50</b> may comprise at least one selected from the group consisting of an etching apparatus, a deposition apparatus, an ion implantation apparatus, an ashing apparatus for removing a photoresist pattern, and a cleaning apparatus.
p-0044Semiconductor device manufacturing methods will be described using the above-described semiconductor device manufacturing equipment <b>100</b>.
p-0045<figref idrefs="DRAWINGS">FIG. 2</figref> is a flowchart illustrating the methods of fabricating a semiconductor device according to an exemplary embodiment, and <figref idrefs="DRAWINGS">FIGS. 3 through 5</figref> are plan views illustrating the methods of fabricating a semiconductor device according to an exemplary embodiment.
p-0046Referring to <figref idrefs="DRAWINGS">FIGS. 2 and 3</figref>, a lower pattern <b>62</b> is formed on a substrate <b>60</b> (S<b>10</b>). The lower pattern <b>62</b> may be formed by the unit process equipment <b>50</b>. The lower pattern <b>62</b> may be formed on an entire surface of the substrate <b>60</b>. The lower pattern <b>62</b> may include various lower overlay patterns which are used in the overlay measuring process. The lower overlay patterns may serve as reference patterns upon measuring the overlay. The lower pattern <b>62</b> may be formed on the substrate <b>60</b> using a lower photoresist pattern as a mask pattern. Subsequently, the lower photoresist pattern on the substrate <b>60</b> may be removed. Moreover, a thin film may be deposited on the lower pattern <b>62</b> using the deposition apparatus of the unit process equipment <b>50</b>.
p-0047Subsequently, referring to <figref idrefs="DRAWINGS">FIGS. 2 and 4</figref>, a photoresist pattern <b>64</b> is formed on the substrate <b>60</b> (S<b>20</b>). The photoresist pattern <b>64</b> is an upper pattern formed on the lower pattern <b>62</b>. The photoresist pattern <b>64</b> may be formed by the spinner equipment <b>20</b> and the exposure equipment <b>30</b>. As described above, the spinner equipment <b>20</b> is capable of coating a photoresist layer on the substrate <b>60</b>. The exposure equipment <b>30</b> is capable of exposing the photoresist layer in accordance with a predetermined-shaped mask pattern. The spinner equipment <b>20</b> is again used to develop the photoresist layer which is exposed by the exposure equipment <b>30</b>. The exposure equipment <b>30</b> is capable of exposing about 100 or more shots with the same shape on the substrate <b>60</b>. Thus, the photoresist pattern <b>64</b> may include a plurality of upper overlay patterns.
p-0048Each of the upper overlay patterns may serve as an overlay pattern used to detect a distortion degree of the photoresist pattern with respect to the corresponding reference pattern upon measuring the overlay. Ideally, when the lower pattern <b>62</b> and the photoresist pattern <b>64</b> are formed using a single reference map in the exposure equipment <b>30</b>, the photoresist pattern <b>64</b> and the lower pattern <b>62</b> may be perfectly aligned (or matched) with each other at all the positions of the substrate <b>60</b>. In fact, however, the photoresist pattern <b>64</b> and the lower pattern <b>62</b> may be misaligned (or unmatched) with each other. In particular, the misalignment values (e.g, the overlay values) between the lower pattern <b>62</b> and the photoresist pattern <b>64</b> may not be uniform at all the positions of the substrate <b>60</b>. That is, the misalignment values between the lower pattern <b>62</b> and the photoresist pattern <b>64</b> may be different from each other. For example, the photoresist pattern <b>64</b> may be formed on the entire surface of the substrate <b>60</b> so as to be deformed in an offset manner by a certain distance from the lower pattern <b>62</b>. Alternatively, the photoresist pattern <b>64</b> may be formed so as to be deformed in a scale manner in proportion by a distance distant from the center of the substrate <b>60</b> in an edge direction. These problems may be due to the natures of the exposure equipment <b>30</b> that exposes the photoresist layer using spherical lens
p-0049As an alternative method of compensating the unmatched formation of the photoresist pattern <b>64</b> over the lower pattern <b>62</b>, there may be used a method of correcting the reference map. That is, it may be required to correct or compensate the initial reference map in order to improve the uniformity of the overlay values and reduce the magnitude of each of the overlay values. The corrected reference map, e.g., the compensated reference map is referred to as a correction map or a compensation map. The exposure equipment <b>30</b> and the spinner equipment <b>20</b> may form a new photoresist pattern using the correction map corrected with an overlap correction value calculated in the overlay measuring equipment <b>40</b>. In consideration of throughput or productivity, the overlay measuring equipment <b>40</b> may perform an overlap measuring process on several shots selected from all shots of the substrate <b>60</b>. The overlay measuring equipment <b>40</b> may calculate a correlation coefficient by applying the overlay measurement values obtained from the overlay measuring process to a polar coordinate function. Then, overlay measuring equipment <b>40</b> may manage the overlay for the entire shots on the substrate <b>60</b> by use of the correlation coefficient. A method of correcting the overlay will be hereinafter described in detail.
p-0050The semiconductor device manufacturing equipment <b>100</b> may generate a reference map, that is, an initial reference map. The reference map may have information relating to at least one position on the substrate <b>60</b>. For example, the reference map may have information relating to a plurality of predetermined positions selected from all positions of the substrate. In an embodiment, each position may indicate a coordinate of any one of all semiconductor chips formed on the substrate. In another embodiment, the predetermined positions may comprise, e.g., 5 positions which are located at a top region, a central region, a bottom region, a left region, and a right region of the substrate. The overlay measuring equipment <b>40</b> receives the reference map from the host computer <b>10</b> or the exposure equipment <b>30</b>. The overlay measuring equipment <b>40</b> may recognize the information regarding the predetermined positions on the substrate <b>60</b>, and the overlay measuring equipment <b>40</b> may move the optical microscopy to the predetermined positions.
p-0051Subsequently, the overlay measuring equipment <b>40</b> performs the overlay measuring process at predetermined positions of the substrate <b>60</b> (S<b>40</b>). That is, the overlay measuring equipment <b>40</b> may measure the overlay values at the predetermined positions. The predetermined positions may be selected by the reference map. At this time, the coordinates of the selected positions may be transferred or stored into the overlay measuring equipment <b>40</b> by an operator or the host computer <b>10</b>. The positions of the reference map may be expressed by the Cartesian coordinate system and the polar coordinates. The overlay measuring equipment <b>40</b> may acquire a measured image indicating the lower pattern <b>62</b> and the photoresist pattern <b>64</b> on the substrate <b>60</b> at every position.
p-0052Subsequently, the overlay measuring equipment <b>40</b> may acquire the overlay measurement value by use of the measured image (S<b>50</b>). The overlay measurement value may correspond to the distorted distance of the lower pattern <b>62</b> and the photoresist pattern <b>64</b> indicated by the measured image. The overlay measurement value may be changed into an X-axis component and a Y-axis component in the Cartesian coordinate system. The overlay measuring equipment <b>40</b> may acquire the plurality of overlay measurement values at all the positions to find out a certain regularity of the overlay measurement values on the entire surface of the substrate <b>60</b>.
p-0053Referring to <figref idrefs="DRAWINGS">FIG. 5</figref>, the overlay measuring equipment <b>40</b> may obtain the plurality of overlay measurement values <b>80</b> at some positions corresponding to about fifteen to about twenty shots <b>70</b> on a single substrate <b>60</b>. Here, the plurality of overlay measurement values <b>80</b> are illustrated to more easily explain the overlay as the values measured using images inside the shots <b>70</b>. Reference Numeral “<b>90</b>” denotes an overlay of the photoresist pattern <b>64</b> formed on the substrate <b>60</b>.
p-0054As described above, the plurality of overlay measurement values <b>80</b> have to be ideally acquired as “0” or a value similar to “0”. In fact, however, the overlay measurement values may be acquired as different values distributed in the overlay measurement. The overlay measuring equipment <b>40</b> may find out the regularity of the plurality of overlay measurement values <b>80</b> to generate the correction map. In an exemplary embodiment, the overlay measuring equipment <b>40</b> may perform an overlay measurement process only at some positions of all the positions in order to generate the correction map. This is because the productivity may be reduced when the overlay measuring process is performed at the positions of all the shots <b>70</b> in the semiconductor device manufacturing process. The regularity of the plurality of overlay measurement values <b>80</b> may be given by the following regressive equation 1. <br />U=AK <Equation 1>
p-0055wherein, “U” represents the plurality of overlay measurement values <b>80</b> and “AK” represents a basis function.
p-0056The basis function may have parameters corresponding to the coordinates of the positions obtained by the plurality of overlay measurement values <b>80</b>. Hereinafter, the parameters correspond to the coordinates of the positions, but may be explained as corresponding to the positions. The parameters may include a radial component (R) and an azimuth angle component (θ). The basis function may include a polar coordinate function in which the radial (R) component and the azimuth angle (θ) component act as independent variables. For example, the polar coordinate function may include a polynomial function (A) and a correlation coefficient (K). Therefore, the polar coordinate function corresponds to the product of the polynomial function (A) and the correlation coefficient (K). The correlation coefficient (K) may be a management parameter of the overlay measuring equipment <b>40</b> for generating the correction map. The overlay measuring equipment <b>40</b> may improve uniformity of the plurality of overlay measurement values <b>80</b> by managing the correlation coefficient (K). When the plurality of overlay measurement values <b>80</b> and the basis function are expressed as a matrix, the correlation coefficient (K) may be calculated by the following equations 2-1 and 2-2. <br />A<sup>T</sup>U=A<sup>T</sup>AK <Equation 2-1><br />A<sup>T</sup>U=K <Equation 2-2>
p-0057wherein, “A” represents a matrix of the polynomial function, and “A<sup>T</sup>” represents a transposed matrix of the polynomial function.
p-0058When the matrix of the polynomial function is orthogonal to the transposed matrix, A<sup>T</sup>A may be a unit matrix. “U” is a matrix of the plurality of overlay measurement values and “K” is a correlation coefficient matrix.
p-0059Accordingly, the overlay measuring equipment <b>40</b> may fit the plurality of overlay measurement values <b>80</b> to the basis function to calculate the correlation coefficients of the basis function for generating the correction map.
p-0060In an embodiment, the polynomial function may include a Zernike polynomial. The Zernike polynomial may be expressed by the product of a radial polynomial component and a trigonometric function component. For example, the Zernike polynomial may be expressed by the following equations 3-1 and 3-2. <br /><i>Z</i><sub>n</sub><sup>−m</sup>(ρ,φ)=<sup>o</sup><i>U</i><sub>n</sub><sup>m</sup>(ρ,φ)=<i>R</i><sub>n</sub><sup>m</sup>(ρ)sin(<i>m</i>φ) <Equation 3-1><br /><i>Z</i><sub>n</sub><sup>m</sup>(ρ,φ)=<sup>e</sup><i>U</i><sub>n</sub><sup>m</sup>(ρ,φ)=<i>R</i><sub>n</sub><sup>m</sup>(ρ)cos(<i>m</i>φ) <Equation 3-2>
p-0061wherein, φ represents an azimuth angle in radians within a range of 0≦φ≦2π and “ρ” represents a radial distance of 0≦ρ≦1.
p-0062The Zernike polynomial uses the radial component and the azimuth angle component as the parameters, and the radial component and the azimuth angle component are orthogonal to each other at each position on the circular substrate <b>60</b> having a normalized radial distance. Thus, it is possible to optimize the uniformity of the plurality of overlay measurement values. The Zernike polynomial may be expressed by even polynomials and odd polynomials. For example, the Zernike polynomial may include the sum of Z<sub>0</sub><sup>0</sup>=1, Z<sub>1</sub><sup>1</sup>=ρ cos φ, Z<sub>1</sub><sup>−1</sup>=ρ sin φ, Z<sub>2</sub><sup>0</sup>=2ρ<sup>2</sup>−1, . . . .
p-0063Accordingly, the plurality of overlay measurement values <b>80</b> may be the product of the Zernike polynomial and the correlation coefficient, as expressed by the following equations 4-1 and 4-2. <br /><i>dx=C</i><sub>0</sub><i>+C</i><sub>2</sub><i>R </i>cos θ+<i>C</i><sub>4</sub><i>R </i>sin θ+<i>C</i><sub>6</sub>(2<i>R</i><sup>2</sup>−1)+ <Equation 4-1><br /><i>dy=C</i><sub>1</sub><i>+C</i><sub>3</sub><i>R </i>cos θ+<i>C</i><sub>5</sub><i>R </i>sin θ+<i>C</i><sub>7</sub>(2<i>R</i><sup>2</sup>−1)+ <Equation 4-2>
p-0064wherein, “dx” and “dy” indicate the overlay measurement values of the Cartesian coordinate system, and “C<sub>0</sub>, C<sub>2</sub>, C<sub>4</sub>, C<sub>6</sub>, . . . ” and “C<sub>1</sub>, C<sub>3</sub>, C<sub>5</sub>, C<sub>7</sub>, . . . ” indicate the correlation coefficients of the X-axis component and the Y-axis component, respectively.
p-0065In an embodiment, “C<sub>0</sub>” and “C<sub>1</sub>” may be correlation coefficients relating to offset deformation. “C<sub>2</sub>” and “C<sub>3</sub>” may be correlation coefficients relating to scale deformation in the X-axis direction. “C<sub>4</sub>” and “C<sub>5</sub>” may be correlation coefficients relating to scale deformation in the Y-axis direction. “C<sub>6</sub>” and “C<sub>7</sub>” may be correlation coefficients relating to high-order deformation. “C<sub>8</sub>, . . . ” and “C<sub>9</sub>, . . . ” higher than “C<sub>6</sub>” and “C<sub>7</sub>” may also be the correlation coefficients relating to high-order deformation.
p-0066Accordingly, the overlay measuring equipment <b>40</b> may improve the uniformity of the plurality of overlay measurement values by mainly managing the correlation coefficients relating to the offset deformation or the scale deformation. The overlay measuring equipment <b>40</b> may further improve the uniformity of the overlay measurement values by managing the correlation coefficients multiplied by the Zernike polynomials in which “m” is 4 or less in the equation 3. For example, the overlay measuring equipment <b>40</b> may multiply the correlation coefficients by 1, R cos θ, R sin θ, (2R<sup>2</sup>−1), R<sup>2 </sup>cos θ, R<sup>2 </sup>sin θ, (3R<sup>3</sup>−2R)cos θ, (3R<sup>3</sup>−2R)sin θ, (6R<sup>4</sup>−6R<sup>2</sup>+1), R<sup>3 </sup>cos 3θ, R<sup>3 </sup>sin 3θ, (4R<sup>4</sup>−3R<sup>2</sup>)cos 2θ, (4R<sup>4</sup>−3R<sup>2</sup>)sin 2θ, (10R<sup>5</sup>−12R<sup>3</sup>+3R)cos θ, (10R<sup>5</sup>−12R<sup>3</sup>+3R)sin θ, 20R<sup>6</sup>−30R<sup>4</sup>+12R<sup>2</sup>−1, R<sup>4 </sup>cos 4θ, R<sup>4 </sup>sin 4θ, (5R<sup>5</sup>+4R<sup>3</sup>)cos 3θ, (5R<sup>5</sup>+4R<sup>3</sup>)sin 3θ, (15R<sup>6</sup>−20R<sup>4</sup>+6R<sup>2</sup>)cos 2θ, (15R<sup>6</sup>−20R<sup>4</sup>+6R<sup>2</sup>)sin 2θ, (35R<sup>7</sup>−60R<sup>5</sup>+30R<sup>3</sup>−4R)cos θ, (35R<sup>7</sup>−60R<sup>5</sup>+30R<sup>3</sup>−4R)sin θ, and 70R<sup>8</sup>−140R<sup>6</sup>+90R<sup>4</sup>−20R<sup>2</sup>+1.
p-0067The plurality of overlay measurement values <b>80</b> may be expressed by matrixes such as the following equations 5-1 and 5-2.
p-0068<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mrow><mrow><mo> </mo><mtable><mtr><mtd><msub><mi>dx</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>5</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>1</mn></mrow><mo>〉</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mrow><mrow><mo> </mo><mtable><mtr><mtd><msub><mi>dy</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>5</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>5</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>2</mn></mrow><mo>〉</mo></mrow></mtd></mtr></mtable></math></maths>
p-0069wherein, “dx<sub>1</sub>” and “dy<sub>1</sub>” may be overlap measurement values measured at positions of “x<sub>1</sub>” and “y<sub>1</sub>” in the Cartesian coordinate system, respectively. “R<sub>1</sub>” and “θ<sub>1</sub>” may correspond to the radial component and the azimuth angle component in the polar coordinate system for indicating the positions of “x<sub>1</sub>” and “y<sub>1</sub>”, respectively. “R<sub>1</sub>” may be a normalized radial distance having a value of 0≦R≦1. Likewise, “dx<sub>2</sub>, dx<sub>3</sub>, . . . , dx<sub>n</sub>” and “dy<sub>e</sub>, dy<sub>3</sub>, . . . , dy<sub>n</sub>” may be overlap measurement values measured at the positions of “x<sub>2</sub>, x<sub>3</sub>, . . . , x<sub>n</sub>” and “y<sub>2</sub>, y<sub>3</sub>, . . . , y<sub>n</sub>”, respectively. “R<sub>2</sub>, R<sub>3</sub>, . . . , R<sub>n</sub>” and “θ<sub>2</sub>, θ<sub>3</sub>, . . . , θ<sub>n</sub>” may also correspond to the radial components and the azimuth angle components in the polar coordinate system for indicating the positions of “x<sub>2</sub>, x<sub>3</sub>, . . . , x<sub>n</sub>” and “y<sub>2</sub>, y<sub>3</sub>, . . . , y<sub>n</sub>”, respectively.
p-0070The matrix of the overlay measurement values may correspond to the product of a correlation coefficient matrix and the Zernike polynomial matrix in which the random positions are parameters. Therefore, the overlay measuring equipment <b>40</b> permits the plurality of overlay measurement values to correspond to the product of a correlation coefficient matrix and the Zernike polynomial matrix (S<b>60</b>).
p-0071The correlation coefficient matrix may be obtained by multiplying the Zernike polynomial transposed matrix by the equations 5-1 and 5-2 on the basis of the equation 2-1 or 2-2. The Zernike polynomial guarantees the orthogonality between the parameters and has no collinearity corresponding to the correlation between individual components of a polynomial. The Zernike polynomial matrix may have “0” in all of the diagonal areas and the non-diagonal areas, when the Zernike polynomial transposed matrix is multiplied. Accordingly, the product of the Zernike polynomial matrix and the Zernike polynomial transposed matrix may be a unit matrix as denoted by the following equation 6-1 or 6-2.
p-0072<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><msup><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo>]</mo></mrow><mi>T</mi></msup><mo></mo><mrow><mo> </mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>dx</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>)</mo></mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>6</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>1</mn></mrow><mo>〉</mo></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo>]</mo></mrow><mi>T</mi></msup><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mrow><mo> </mo><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>dy</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>…</mi></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>)</mo></mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>5</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>6</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>2</mn></mrow><mo>〉</mo></mrow></mtd></mtr></mtable></math></maths>
p-0073In the equations 6-1 and 6-2, the product of the Zernike polynomial matrix and the Zernike polynomial transposed matrix may be expressed as a unit matrix with “ones” on the main diagonal or may be expressed as a unit matrix multiplied by a constant. When the Zernike polynomial transposed matrix is calculated and the unit matrix is removed, the equations 6-1 and 6-2 may be changed into the following equations 7-1 and 7-2, respectively.
p-0074<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>…</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>dx</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dx</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mo> </mo><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>4</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>7</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>1</mn></mrow><mo>〉</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>…</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>cos</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>R</mi><mn>1</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>2</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow><msub><mi>R</mi><mn>3</mn></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mn>3</mn></msub></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><msub><mi>R</mi><mi>n</mi></msub><mo></mo><mi>sin</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>θ</mi><mi>n</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mn>3</mn><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd><mtd><mi>…</mi></mtd><mtd><mrow><mrow><mn>2</mn><mo></mo><msubsup><mi>R</mi><mi>n</mi><mn>2</mn></msubsup></mrow><mo>-</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd><mtd><mi>⋮</mi></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>dy</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>dy</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo> </mo><mo> </mo></mrow><mo>[</mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo></mo><mtable><mtr><mtd><msub><mi>C</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msub><mi>C</mi><mn>5</mn></msub></mtd></mtr><mtr><mtd><mi>⋮</mi></mtd></mtr><mtr><mtd><msub><mi>C</mi><mrow><mrow><mn>2</mn><mo></mo><mi>n</mi></mrow><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo></mo><mstyle><mspace width="0.em" height="0.ex" /></mstyle><mo>]</mo></mrow></mrow></mtd><mtd><mrow><mo>〈</mo><mrow><mi>Equation</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mn>7</mn><mo></mo><mstyle><mtext>-</mtext></mstyle><mo></mo><mn>2</mn></mrow><mo>〉</mo></mrow></mtd></mtr></mtable></math></maths>
p-0075As can be seen from the equations 6 and 7, the correlation coefficient matrix may be expressed by the Zernike polynomial transposed matrix (A<sup>T</sup>) and the matrix (U) of the plurality of overlay measurement values. For example, the correlation coefficient “C<sub>0</sub>” may correspond to an offset deformation value obtained by adding all the plurality of overlay measurement values <b>80</b> corresponding to “dx<sub>1</sub>, dx<sub>2</sub>, dx<sub>3</sub>, . . . , dx<sub>n</sub>”. Similarly, the correlation coefficient “C<sub>1</sub>” may correspond to an offset deformation value obtained by adding all the plurality of overlay measurement values <b>80</b> corresponding to “dy<sub>1</sub>, dy<sub>2</sub>, dy<sub>3</sub>, . . . , dy<sub>n</sub>”. The correlation coefficients “C<sub>2</sub>” and “C<sub>3</sub>” may correspond to scale deformation values obtained by adding all the individual products of the X-axis distances at the positions of (R<sub>1</sub>, θ<sub>1</sub>), (R<sub>2</sub>, θ<sub>2</sub>), (R<sub>3</sub>, θ<sub>3</sub>), . . . , (R<sub>n</sub>, θ<sub>n</sub>) and the plurality of overlay measurement values <b>80</b>. The correlation coefficients “C<sub>4</sub>” and “C<sub>5</sub>” may correspond to scale deformation values obtained by adding all the individual products of the Y-axis distances at the positions of (R<sub>1</sub>, θ<sub>1</sub>), (R<sub>2</sub>, θ<sub>2</sub>), (R<sub>3</sub>, θ<sub>3</sub>), . . . , (R<sub>n</sub>, θ<sub>n</sub>) and the plurality of overlay measurement values <b>80</b>. Likewise, the correlation coefficients “C<sub>6</sub>” and “C<sub>7</sub>” may correspond to high-order deformation values. Accordingly, the overlay measuring equipment <b>40</b> may calculate the correlation coefficients at the positions on the substrate <b>60</b> by use of the plurality of overlay measurement values (S<b>70</b>).
p-0076The overlay measuring equipment <b>40</b> may calculate overlay correction value using the correlation coefficient matrix (S<b>80</b>). The overlay correction value may be calculated by the product of the correlation coefficient matrix and the Zernike polynomial matrix in which all of the individual positions at which the overlay pattern is formed in the entire substrate <b>60</b> are parameters. For example, the overlay measuring equipment <b>40</b> may calculate the overlay correction value in all the shots <b>70</b> by multiplying the correlation coefficient matrix by the Zernike polynomial matrix in which the position of each shot of a total of about 100 or more shots <b>70</b> on the substrate <b>60</b> is the parameter. The overlay measuring equipment <b>40</b> may correct the reference map by use of the overlay correction value calculated in each shot <b>70</b>. Accordingly, the overlay measuring equipment <b>40</b> may update the reference map using the overlay correction value, thereby generating the correction map (S<b>90</b>).
p-0077The exposure equipment <b>30</b> may receive information of the correction map from the overlay measuring equipment <b>40</b> via the host computer <b>10</b> to perform a corrected (or a compensated) exposure process (S<b>20</b>). The exposure equipment <b>30</b> or the host computer <b>10</b> may also generate the correction map by use of the correlation coefficient matrix and the overlay correction value. Subsequently, the overlay measuring equipment <b>40</b> may repeatedly execute the overlay measuring process in a cycle by resetting the reference map to the correction map. When a certain period expires or the activation of the overlay measuring equipment <b>40</b> stops, the overlay measuring process may end.
p-0078As a consequence, the method of correcting an overlay according to an exemplary embodiment is capable of calculating the overlay correction value improved in the uniformity of the plurality of overlay measurement values by using the Zernike polynomials of the polar coordinate function. Moreover, it is possible to improve or maximize the reliability of the overlay process, since the uniformity of the overlay correction value may be ensured by permitting the plurality of overlay measurement value to correspond to the polar coordinate function.
p-0079As described above, according to the exemplary embodiments, it is possible to calculate the overlay correction value improved in the distribution of the plurality of overlay measurement values by using the Zernike polynomials of the polar coordinate function.
p-0080Moreover, it is possible to improve or maximize the reliability in the process of correcting the overlay, since the overlay measuring equipment may generate the correction map from the reference map by using the correlation coefficient multiplied by the Zernike polynomials.
p-0081Exemplary embodiments have been disclosed herein, and although specific terms are employed, they are used and are to be interpreted in a generic and descriptive sense only and not for purpose of limitation. Accordingly, it will be apparent to those of ordinary skill in the art that various substitution, modifications and changes may be thereto without departing from the scope and spirit of the invention as set forth in the following claims. The above-disclosed subject matter is to be considered illustrative and not restrictive.
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Numbers
- Publication
- 08773153
- Application
- 13029633
Titles
- English
- Method of correcting overlay and semiconductor device manufacturing method using the same
Patent term adjustment
- A delay
- +149 daysthe office missed an examination deadline
- Applicant delay
- −9 days
- Net adjustment
- 140 days
Classification
- CPC, 8
- G03F7/70633
- G03F7/7085
- G03F7/70683
- G03F7/706835
- G03F7/706845
- G03F9/7015
- G03F9/7019
- G03F7/70516
- IPC, 1
- G01R31 00